'Clear Sharp' doesn't seem to be working
There are no sharp edges on the mesh. What you are observing there is geometry obstructing itself or forming some unexpected shadows because of the way it is triangulated. When rendered, the mesh is triangulated and some faces that may look smooth when you rotate them in the 3d view get divided into triangles that obstruct each other looking at them at some angles or produce unexpected shadows: 

 

There is nothing to do about it - the geometry is just not detailed enough to be rendered nicely, you do need to use Subsurf modifier or modify the geometry some other way.